Posts Tagged ‘a polyformist’s toolkit’

(Tagging under “A Polyformist’s Toolkit”, as I feel that series ought to have an entry on coloring, and this more or less says what I have to say about that.)

At Gathering for Gardner 11 in 2014, I gave a talk about crossed stick puzzles. It was the obvious thing to talk about, since I had been making a lot of interesting discoveries in that area. Unfortunately there was too much good stuff, and I couldn’t bear to trim very much of it out, so I made the classic mistake of going over on time and having to rush the last slides. (G4G talks are generally limited to 6 minutes.) When I looking for a subject for this year’s talk, there was nothing I felt an urgent desire to talk about. This would be the 12th Gathering for Gardner, and there is a tradition that using the number of the current Gathering, either in your talk or your exchange gift, is worth a few style points. Since I’m a polyformist, and Gardner famously popularized the twelve pentominoes, revisiting some of my pentomino coloring material seemed reasonable.

Finding interesting map colorings is a nice puzzle that we can layer on top of a tiling problem. A famous theorem states that all planar maps can be colored with four colors so no two regions of the same color touch. Since this can always be done, and fairly easily for small maps like pentomino tilings, we’ll want some properties of colorings that are more of a challenge to find. I know of three good ones:

Three-colorability. Sometimes we only need three colors rather than four. For sufficiently contrived sets of tiles we might only need two, but for typical problems that won’t work.

Strict coloring. For most purposes, (like the Four Color Theorem) we allow regions of the same color to touch at a vertex. If we do not allow same colored regions to touch at a vertex, we call the legal colorings strict. Notice that a 3-coloring of polyominoes is strict if and only if it contains no “crossroads”, i.e. corners where four pieces meet.

Color balance. If the number of regions of each color is equal, a coloring may be considered balanced. Conveniently, 3 and 4 are both divisors of 12, so we can have balanced 3-colorings and 4-colorings of pentomino tilings.

The above information would make up the introduction to my talk. It would also, suitably unpacked and with examples, take up most of the alloted time. That left little enough room to show off nifty discoveries. So whatever nifty discoveries I did show would serve the talk best if they could illustrate the above concepts without adding too many new ones. One that stood out was this simultaneous 3- and 4-coloring with a complete set of color combinations, discovered by Günter Stertenbrink in 2001 in response to a query I made on the Polyforms list:

This is the unique pentomino tiling of a 6×10 rectangle with this property where the colorings are strict. I used it to illustrate 3- vs. 4-coloring by showing the component colorings first, before showing how they combine. To my astonishment, the audience at G4G12 applauded the slide with the combined colorings. I mean I think it’s pretty cool, but I consider it rather old material.

I still wanted one more nifty thing to show off, and while my page on pentomino colorings had several more nifty things, none of them hewed close to the introductory material, and the clever problem involving overlapping colored tilings that I was looking at didn’t seem very promising. Setting that aside, I wrote some code to get counts of the tilings of the 6×10 rectangle with various types of colorings. That gave me the following table:

Total

Balanced

4-colorable, non-strict

2339

2338

4-colorable, strict

2339

2320

3-colorable, non-strict

1022

697

3-colorable, strict

94

53

What stood out to me was the 2338 tilings with balanced colorings. Since there are 2339 tilings in total, that meant that there was exactly one tiling with no balanced coloring:

Notice that the F pentomino on the left borders eight of the other pentominoes, and the remaining three border each other, so there can be at most two pentominoes with the color chosen for the F, and no balanced coloring can exist. A unique saddest tiling balancing out the unique happiest tiling was exactly what my talk needed. Now it had symmetry, and a cohesive shape. Having important examples all using the 6×10 rectangle removed the extraneous consideration of what different tiling problems were out there, and helped to narrow the focus to just the coloring problem. Anyway, I don’t want to go on any more about how awesome of a talk it was, (especially because video of it may eventually go up on the internet, which would show how non-awesome my delivery was) but it was my first G4G talk that I was actually proud of. The slides for the talk are here.

One thing I’m curious about that I didn’t mention in the talk: has anyone else found the saddest tiling before me? Looking through old Polyform list emails, I found that Mr. Stertenbrink enumerated the 3-colorable tilings of various types (essentially, the bottom half of the table above) but not the 4-colorable tilings. From the perspective of looking for the “best” colorings, it makes sense to focus on the 3-colorable tilings, but it meant missing an interesting “worst” coloring.

In polyomino puzzles, we would frequently like to tile the simplest shape possible, and a rectangle usually seems to fit the bill. But sometimes a rectangle isn’t possible. For example, we can never make a 4×5 rectangle with the five tetrominoes. One way to prove this is with a checkerboard parity argument. Four of the 5 tetrominoes must always occupy even numbers of both black and white squares if they are placed on a checkerboard. The T tetromino must occupy odd numbers of each color. Therefore a rectangle must have odd numbers of each color, but any rectangle of size 20 will have colors evenly divided, 10 and 10. A similar argument can be made to show that the 35 hexominoes cannot tile a rectangle.

This will never work…

Rather than give up and accept that we’ll need to find a less elegant shape to tile, we have another option. If we wrap the edges of a 5×4 rectangle around to form a cylinder, (so that the cylinder is 4 squares tall and 5 squares in circumference) tiling is once again possible. To see why this might be so, imagine that you are coloring the squares as in a checkerboard. Once you got back around to where you began, you would find that in order to continue the pattern, you would need to use the opposite colors from those you already used. Note that this would not work if you wrapped the rectangle in the other direction; because the other side has even length, the checkering colors remain consistent.

…until we wrap the rectangle into a cylinder.

There is a video by Edo Timmermans showing how a tetromino cylinder can be made with toy magnets. He claims that there are seven distinct tilings of a cylinder with the tetrominoes, and poses an interesting puzzle involving them. A commercially produced cylindrical polyomino puzzle is Logiq Tower, designed by Marko Pavlović, which uses wooden pentomino-based pieces that form a cylinder together with some other pieces. Because these pieces are inflexible, they lack some of the allowable symmetry actions of free pentominoes.

A cylinder isn’t our only option. We could give the rectangle a half-twist before connecting the ends; this gives us a Möbius strip. We could also connect both pairs of sides instead of one; this gives a structure that is topologically equivalent to a torus or doughnut. And then we could add twists to that— well, at this point it would be nice to be systematic so we can be sure that we’ve found all of the possibilities. One thing to note is that adding more twists doesn’t actually give us more possibilities. A strip with two twists will have exactly the same tilings as a strip with no twists, and in general, a strip with an even number of half-twists will have the same tilings as the no-twist strip, and a strip with an odd number of half-twists will have the same tilings as the Möbius strip. So for each dimension, we have three options: no connection, connection without a twist, and connection with a half-twist. This gives us the following matrix of possibilities:
Only six possibilities here, not nine, because the ones in the lower left are equivalent to the ones across the main diagonal from them. Note that the Möbius strip, Klein bottle, and projective plane are nonorientable surfaces, which means that they effectively have only one side.

An important consideration when working with these is that one-sided polyominoes don’t exist on nonorientable surfaces. With one-sided polyominoes, translation is allowed, but reflection isn’t. However, on a non-orientable surface, translating far enough leaves an object in a reflected state.

Another consideration is that coloring is harder when we leave the plane behind. On the plane, we have a theorem stating that we never need to use more than four colors to make all of the tiles differ in color from all of their neighbors. On a torus, this may require seven colors. In 2001, Roger Phillips found 18 heptominoes that could tile a 7×7 torus, and sent these tilings to MathPuzzle.com. Here’s one:

Depending on the dimensions of the torus, it may be possible for a polyomino to wrap around and touch itself. In a strict sense, this makes any coloring impossible, since we don’t let tiles of the same color touch. However, we can follow a looser standard, and allow self-touching polyominoes in our colored tilings. Patrick Hamlyn found a 3-coloring of a tiling of the 35 hexominoes in 7 3×10 tori using this scheme in 2003:

This problem has no solutions if the tori are replaced with rectangles or cylinders.

Problems #31-37:
Though it seems like a pretty basic problem, if anyone has counted the number of pentomino tilings of cylinders, I am not aware of it. Wrapping the short sides of the 3×20 together should not give any solutions beyond the two obtained by wrapping the solutions on the 3×20 rectangle. That leaves the 3×20 wrapped the other way, and both ways of wrapping the 4×15, the 5×12, and the 6×10 rectangles.

Problems #38-40: Find the solution counts for the 4×15, 5×12, and 6×10 tori. I don’t know if these are all computationally tractable, but I can hope. (The 3×20 will be the same as the 3×20 cylinder with long sides wrapped together.)

Even more possibilities for tiling become available when you choose parallelograms with diagonal sides to wrap around, but this post is long enough, so that will have to be a matter for another post.

It lately occurred to me that there are concepts that I use (and see used by others) in creating variations on polyform puzzles that I haven’t seen explained very thoroughly, and it might be helpful if I used this space for just that purpose.

Some polyomino puzzles using symmetry variations

The first of these is the use of different kinds of symmetry in defining the set of pieces used in a puzzle. (I touched on this in my post on rectangular-cell pentominoes.) Normally, all combinations of rotations, translations, and reflections of a polyomino in a grid are considered to be equivalent. Leaving aside translations for the moment, the possible rotations and reflections of a polyomino are equivalent to the group of symmetries of a square. We can find variations on polyominoes by restricting the allowed symmetries to subgroups of that group. For example, the one-sided polyominoes are the result of allowing only rotations, not reflections. Rhombic cell pentominoes (which Kadon sells) allow 180° rotations, plus diagonal reflections. My Agincourt puzzle allows only reflections over vertical axes, assuming that the arrows are pointing vertically. Notice that it doesn’t matter which direction the arrows point as long as they point in the same direction; this suggests that what we are interested in isn’t symmetry subgroups per se, but classes of subgroups where two subgroups that are related to each other by symmetries of the square are equivalent.

What are all of the possible variations with different allowed transformations? We can generate a representative subgroup of every class by using some combination of reflection over a particular axis parallel to the grid, a particular diagonal axis, and 90° and 180° rotations. Here’s a chart of the symmetry variations this produces.

Polyomino Type

Reflection

Rotation

# of Symmetries

Free

Either

90°

8

Parallel (a.k.a. Rectangular)

y axis

180°

4

Diagonal (a.k.a. Rhombic)

x=y

180°

4

One-sided

None

90°

4

Oriented Parallel

y axis

None

2

Oriented Diagonal

x=y

None

2

Polar One-sided

None

180°

2

Fixed

None

None

1

I chose the above terminology for the types (after keeping “free”, “one-sided”, and “fixed” as established terms) in order to build in some helpful mnemonics. The types with four symmetries have short names. The types with two symmetries have longer names based on the names of the types whose symmetry groups their symmetries are subgroups of. The odd duck here is “polar one-sided”, which is a subgroup of all of the larger symmetry groups, but putting “one-sided” in its name makes the types with two symmetries nicely echo the names of those with four.

Here’s a chart of the number of polyominoes of each type for a given size:

(The odd entries for the polar one-sided polyominoes track those for the oriented parallel polyominoes exactly for several terms, before eventually diverging. There are 4998 9-ominoes for both, and 67792 polar one-sided, and 67791 oriented parallel 11-ominoes. It seems unlikely that this is a coincidence. Does anyone know why this occurs?)

These types can be realized geometrically by replacing square cells in a planar tiling with cells with the appropriate symmetry. Another way they can be realized is by keeping the cells square and marking them with a figure with the appropriate symmetry. This is essentially what I did by cutting arrow shaped holes in the Agincourt pieces. The latter method allows the possibility of mixing different symmetry types in the same tiling. I don’t believe I’ve seen such a problem before, so let me be the first to fill what may be a much needed gap:

Problem #28: Tile a 6×6 square with the oriented parallel, oriented diagonal, and polar one-sided trominoes. No tromino should touch another of the same type.

With these symmetry subgroup based polyform variations in mind, any type of polyform on a square grid can be transformed into an entire family of polyforms. In particular, polysticks would reward exploration in this light, which does not seem to have occurred yet. A similar analysis to the one above can be made for symmetry based variations of polyiamonds and polyhexes. Bringing translation symmetry subgroups into the picture leads to things like checkered polyominoes. I may get to these in later posts; this one was getting long enough that I needed to wrap it up.

I should note that Peter Esser’s pages on polyforms cover these variations, and that his polyomino solver program can work with any of the 8 symmetry types (but not with mixed types.) (It is, sadly, a Windows binary, but I’ve been able to make it work under Wine on Linux.)

I have had a lifelong interest in recreational mathematics, especially polyomino problems. I’ve produced some puzzles in laser-cut plastic, which I sell via this very site. I’ve also dabbled in writing interactive fiction.